Local well-posedness and blow-up in the energy space for a class of $ {L}^{2}$ critical dispersion generalized Benjamin–Ono equations

Kenig, C.E.; Martel, Y.; Robbiano, L.

Local well-posedness and blow-up in the energy space for a class of

L^{2}

critical dispersion generalized Benjamin–Ono equations

Kenig, C.E. ; Martel, Y. ; Robbiano, L.

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011), p. 853-887 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

Nous considérons une famille dʼéquations de Benjamin–Ono à dispersion généralisée (dgBO) $u_{t} - \partial_{x} {| D |}^{α} u + {| u |}^{2 α} \partial_{x} u = 0, (t, x) \in ℝ \times ℝ,$ où $\hat{{| D |}^{α} u} = {| ξ |}^{α} \hat{u}$ et $1 ⩽ α ⩽ 2$ . Ces équations sont critiques par rapport à la norme $L^{2}$ et à lʼexistence globale et peuvent être vues comme des interpolations entre lʼéquation de Benjamin–Ono généralisée critique ( $α = 1$ ) et lʼéquation de Korteweg–de Vries généralisée critique ( $α = 2$ ).Dʼabord, nous montrons le caractère bien posé de ces équations dans lʼespace dʼénergie pour $1 < α < 2$ , étendant les résultats de Kenig et al. (1991, 1993) [13,14] pour les équations de Korteweg–de Vries généralisées.Ensuite, nous étudions le phénomène dʼexplosion dans lʼesprit de Martel et Merle (2000) [19] et Merle (2001) [22] concernant lʼéquation de gKdV critique, en étudiant les propriétés de rigidité du flot de dgBO dans un voisinage des solitons. Nous montrons que pour α proche de 2, les solutions dʼénergie négative proches des solitons explosent en temps fini ou infini dans lʼespace dʼénergie $H^{\frac{α}{2}}$ .La preuve de ce résultat dʼexplosion est basée dʼune part sur lʼadaptation à dgBO de résultats de monotonie de normes $L^{2}$ locales par des méthodes dʼopérateurs pseudo-differentiels et dʼautre part sur des arguments de perturbation pour obtenir des propriétés structurelles du flot linéarisé autour des solitons lorsque lʼéquation est proche de gKdV.

We consider a family of dispersion generalized Benjamin–Ono equations (dgBO) $u_{t} - \partial_{x} {| D |}^{α} u + {| u |}^{2 α} \partial_{x} u = 0, (t, x) \in ℝ \times ℝ,$ where $\hat{{| D |}^{α} u} = {| ξ |}^{α} \hat{u}$ and $1 ⩽ α ⩽ 2$ . These equations are critical with respect to the $L^{2}$ norm and global existence and interpolate between the modified BO equation ( $α = 1$ ) and the critical gKdV equation ( $α = 2$ ).First, we prove local well-posedness in the energy space for $1 < α < 2$ , extending results in Kenig et al. (1991, 1993) [13,14] for the generalized KdV equations.Second, we address the blow-up problem in the spirit of Martel and Merle (2000) [19] and Merle (2001) [22] concerning the critical gKdV equation, by studying rigidity properties of the dgBO flow in a neighborhood of the solitons. We prove that for α close to 2, solutions of negative energy close to solitons blow up in finite or infinite time in the energy space $H^{\frac{α}{2}}$ .The blow-up proof requires both extensions to dgBO of monotonicity results for local $L^{2}$ norms by pseudo-differential operator tools and perturbative arguments close to the gKdV case to obtain structural properties of the linearized flow around solitons.

Publié le : 2011-01-01

Zbl 1230.35102

DOI : https://doi.org/10.1016/j.anihpc.2011.06.005

@article{AIHPC_2011__28_6_853_0,
     author = {Kenig, C.E. and Martel, Y. and Robbiano, L.},
     title = {Local well-posedness and blow-up in the energy space for a class of $ {L}^{2}$ critical dispersion generalized Benjamin--Ono equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {28},
     year = {2011},
     pages = {853-887},
     doi = {10.1016/j.anihpc.2011.06.005},
     zbl = {1230.35102},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2011__28_6_853_0}
}

Kenig, C.E.; Martel, Y.; Robbiano, L. Local well-posedness and blow-up in the energy space for a class of $ {L}^{2}$ critical dispersion generalized Benjamin–Ono equations. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) pp. 853-887. doi : 10.1016/j.anihpc.2011.06.005. http://gdmltest.u-ga.fr/item/AIHPC_2011__28_6_853_0/

Bibliographie

[1] C.J. Amick, J.F. Toland, Uniqueness and related analytic properties for the Benjamin–Ono equation—a nonlinear Neumann problem in the plane, Acta Math. 167 (1991), 107-126 | MR 1111746 | Zbl 0755.35108

[2] C.J. Amick, J.F. Toland, Uniqueness of Benjaminʼs solitary-wave solution of the Benjamin–Ono equation, IMA J. Appl. Math. 46 (1991), 21-28 | MR 1106251 | Zbl 0735.35105

[3] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245-1260 | MR 2354493 | Zbl 1143.26002

[4] S. Cui, C.E. Kenig, Weak continuity of the flow map for the Benjamin–Ono equation on the line, J. Fourier Anal. Appl. 16 (2010), 1021-1052 | MR 2737768 | Zbl 1223.35272

[5] R.L. Frank, E. Lenzmann, Uniqueness and nondegeneracy of ground states for ${(- Δ)}^{s} Q + Q - Q^{α + 1} = 0$ in $ℝ$ , arXiv:1009.4042v1

[6] B.V. Gnedenko, A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison–Wesley Publishing Company (1954) | MR 62975 | Zbl 0056.36001

[7] O. Goubet, L. Molinet, Global weak attractor for weakly damped nonlinear Schrödinger equations in $L^{2} (ℝ)$ , Nonlinear Anal. 71 (2009), 317-320 | MR 2518038 | Zbl 1170.35534

[8] P.R. Halmos, V.S. Sunder, Bounded Integral Operators on $L^{2}$ Spaces, Ergeb. Math. Grenzgeb. vol. 96, Springer-Verlag, Berlin (1978) | MR 517709 | Zbl 0389.47001

[9] Lars Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudodifferential Operators, Grundlehren Math. Wiss. vol. 274, Springer-Verlag, Berlin (1985) | MR 781536 | Zbl 0601.35001

[10] T. Kato, On the Cauchy problem for the (generalized) Korteweg–de Vries equation, Studies in Applied Mathematics, Adv. Math. Suppl. Stud. vol. 8, Academic Press, New York (1983), 93-128 | MR 759907

[11] C.E. Kenig, Y. Martel, Asymptotic stability of solitons for the Benjamin–Ono equation, Rev. Mat. Iberoam. 25 (2009), 909-970 | MR 2590690 | Zbl 1247.35133

[12] C.E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69 | MR 1101221 | Zbl 0738.35022

[13] C.E. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the Korteweg–de Vries equation, J. Amer. Math. Soc. 4 (1991), 323-347 | MR 1086966 | Zbl 0737.35102

[14] C.E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), 527-620 | MR 1211741 | Zbl 0808.35128

[15] C.E. Kenig, H. Takaoka, Global wellposedness of the modified Benjamin–Ono equation with initial data in $H^{1 / 2}$ , Int. Math. Res. Not. (2006) | MR 2219229 | Zbl 1140.35386

[16] P.-L. Lions, The concentration–compactness principle in the calculus of variations: the locally compact case. Parts 1 and 2, Ann. Inst. H. Poincaré Non Linéaire 1 (1984), 109-145 | Numdam | Numdam | MR 778970 | Zbl 0541.49009

[17] Y. Martel, Linear problems related to asymptotic stability of solitons of the generalized KdV equations, SIAM J. Math. Anal. 38 (2006), 759-781 | MR 2262941 | Zbl 1126.35055

[18] Y. Martel, F. Merle, Instability of solitons for the critical generalized Korteweg–de Vries equation, Geom. Funct. Anal. 11 (2001), 74-123 | MR 1829643 | Zbl 0985.35071

[19] Y. Martel, F. Merle, A Liouville theorem for the critical generalized Korteweg–de Vries equation, J. Math. Pures Appl. 79 (2000), 339-425 | MR 1753061 | Zbl 0963.37058

[20] Y. Martel, F. Merle, Stability of the blow up profile and lower bounds on the blow up rate for the critical generalized KdV equation, Ann. of Math. 155 (2002), 235-280 | MR 1888800 | Zbl 1005.35081

[21] Y. Martel, F. Merle, Blow up in finite time and dynamics of blow up solutions for the $L^{2}$ -critical generalized KdV equation, J. Amer. Math. Soc. 15 (2002), 617-664 | MR 1896235 | Zbl 0996.35064

[22] F. Merle, Existence of blow-up solutions in the energy space for the critical generalized Korteweg–de Vries equation, J. Amer. Math. Soc. 14 (2001), 555-578 | MR 1824989 | Zbl 0970.35128

[23] F. Merle, P. Raphaël, On universality of blow up profile for $L^{2}$ critical nonlinear Schrödinger equation, Invent. Math. 156 (2004), 565-672 | MR 2061329 | Zbl 1067.35110

[24] F. Merle, P. Raphaël, Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), 37-90 | MR 2169042 | Zbl 1075.35077

[25] F. Merle, P. Raphaël, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys. 253 (2005), 675-704 | MR 2116733 | Zbl 1062.35137

[26] L. Molinet, F. Ribaud, Well-posedness results for the generalized Benjamin–Ono equation with small initial data, J. Math. Pures Appl. 83 (2004), 277-311 | MR 2038121 | Zbl 1084.35094

[27] M. Reed, B. Simon, Methods of Modern Mathematical Physics IV. Analysis of Operators, Academic Press, New York, San Francisco, London (1978) | MR 493421 | Zbl 0401.47001

[28] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1983), 567-576 | MR 691044 | Zbl 0527.35023

[29] M.I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472-491 | MR 783974 | Zbl 0583.35028

[30] M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), 51-68 | MR 820338 | Zbl 0594.35005

[31] M.I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations 12 (1987), 1133-1173 | MR 886343 | Zbl 0657.73040

[32] M.I. Weinstein, Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities, J. Differential Equations 69 (1987), 192-203 | MR 899159 | Zbl 0636.35015

[33] M. Yamazato, Unimodality of infinitely divisible distribution functions of class L, Ann. Probab. 6 (1978), 523-531 | MR 482941 | Zbl 0394.60017